# Comparing Geometric Discord and Negativity for Bipartite States

## TL;DR

This paper investigates the relationship between geometric discord and negativity in bipartite quantum states, providing counterexamples to a previously conjectured inequality and establishing bounds for their difference across various dimensions.

## Contribution

The paper presents analytic families of states violating the geometric discord-negativity relation and derives bounds for their difference in arbitrary bipartite systems.

## Key findings

- Counterexamples for $ ext{C}^2 imes ext{C}^3$ states violating the conjecture.
- Bounds for $ ext{N}^2 - ext{D}$ in $ ext{C}^m imes ext{C}^n$ systems.
- The conjecture does not hold universally beyond two-qubit states.

## Abstract

The geometric discord $\mathcal{D}$ of a state is a measure of the quantumness of the state and the negativity $\mathcal{N}$ is a measure of the entanglement of a state. It was proved by D. Girolami and G. Adesso that for states on $\mathbb{C}^2\otimes\mathbb{C}^2$, the geometric discord is always greater than or equal to the square of the negativity and conjectured that this holds in general. S. Rana and P. Parashar showed that this relation does not hold for all states on $\mathbb{C}^2\otimes\mathbb{C}^n$ for $n>2$. We provide several analytic families of states on $\mathbb{C}^2\otimes\mathbb{C}^3$ violating this relation. Certain upper and lower bounds for $\mathcal{N}^2-\mathcal{D}$ are obtained for states on $\mathbb{C}^m\otimes\mathbb{C}^n$ for any $m, n\in\mathbb{N}$.